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44
A Recognition Algorithm for Classical Groups over Finite Fields
 Proc. London Math. Soc
, 1998
"... 2. Classical groups and primitive prime divisors...... 121 3. Generic and nongeneric parameters........ 123 4. Groups with two different primitive prime divisor elements... 126 ..."
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2. Classical groups and primitive prime divisors...... 121 3. Generic and nongeneric parameters........ 123 4. Groups with two different primitive prime divisor elements... 126
A Constructive Recognition Algorithm for the Special Linear Group
"... In the rst part of this note we present an algorithm to recognise constructively the special linear group. In the second part we give timings and examples. 1 Introduction It seems possible, using Aschbacher's celebrated analysis of subgroups of classical groups [5], to develop algorithms that ..."
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In the rst part of this note we present an algorithm to recognise constructively the special linear group. In the second part we give timings and examples. 1 Introduction It seems possible, using Aschbacher's celebrated analysis of subgroups of classical groups [5], to develop algorithms that will answer basic questions about the group G generated by a subset X of GL(d; q), for modest values of d and q, as is already possible for permutation groups. The best strategy may involve trying to recognise very large subgroups of GL(d; q) by special techniques. In the case of permutation groups, special techniques are used to recognise the alternating and symmetric groups. This is done by making a random search for elements of a certain cycle type. If such elements are found in a primitive group, the group is known to contain the alternating group. If no such elements are found after a suÆciently long search, one proceeds with the expectation that one is dealing with a smaller group. For li...
Recognising Tensor Products of Matrix Groups
 Internat. J. Algebra Comput
, 1997
"... As a contribution to the project for recognising matrix groups defined over finite fields, we describe an algorithm for deciding whether or not the natural module for such a matrix group can be decomposed into a nontrivial tensor product. In the affirmative case, a tensor decomposition is return ..."
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As a contribution to the project for recognising matrix groups defined over finite fields, we describe an algorithm for deciding whether or not the natural module for such a matrix group can be decomposed into a nontrivial tensor product. In the affirmative case, a tensor decomposition is returned. As one component, we develop algorithms to compute plocal subgroups of a matrix group. 1991 Mathematics Subject Classification (Amer. Math. Soc.): 20C20, 20C40. 1 Introduction In LeedhamGreen & O'Brien (1996), we give an internal description of tensor decompositions of finitedimensional vector spaces. We do this by constructing a family of projective geometries whose flats are certain subspaces of a vector space V , and showing that there is a onetoone correspondence between this family of projective geometries and the set of equivalence classes of tensor decompositions of V . The object of this paper is to exploit the geometrical approach presented there, together with some o...
Algorithms for Matrix Groups and the Tits Alternative
 Proc. 36th IEEE FOCS
, 1999
"... l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals ..."
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l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals and Babai, who give a Las Vegas algorithm for the case of finite groups, as well as recent work of Babai, Beals, Cai, Ivanyos, and Luks, who give a deterministic algorithm for the case of abelian groups. # 1999 Academic Press Article ID jcss.1998.1614, available online at http:##www.idealibrary.com on 260 00220000#99 #30.00 Copyright # 1999 by Academic Press All rights of reproduction in any form reserved. * Research conducted while visiting IAS and DIMACS and supported in part by an NSF Mathematical Sciences
Prime power graphs for groups of Lie type
 JOURNAL OF ALGEBRA
, 2002
"... We associate a weighted graph (G) to each nite simple group G of Lie type. We show that, with an explicit list of exceptions, (G) determines G up to isomorphism, and for these exceptions, (G) nevertheless determines the characteristic of G. This result was motivated by algorithmic considerations. ..."
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We associate a weighted graph (G) to each nite simple group G of Lie type. We show that, with an explicit list of exceptions, (G) determines G up to isomorphism, and for these exceptions, (G) nevertheless determines the characteristic of G. This result was motivated by algorithmic considerations. We prove that for any nite simple group G of Lie type, input as a black box group with an oracle to compute the orders of group elements, (G) and the characteristic of G can be computed by a Monte Carlo algorithm in time polynomial in the input length. The characteristic is needed as part of the input in a previous constructive recognition algorithm for G.
Polynomial Time Algorithms for Modules Over Finite Dimensional Algebras
, 1997
"... We present polynomial time algorithms for some fundamental tasks from representation theory of finite dimensional algebras. These involve testing (and constructing) isomorphisms of modules as well as expressing of modules as direct sums of indecomposable modules. Over number fields the latter task s ..."
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We present polynomial time algorithms for some fundamental tasks from representation theory of finite dimensional algebras. These involve testing (and constructing) isomorphisms of modules as well as expressing of modules as direct sums of indecomposable modules. Over number fields the latter task seems to be difficult, therefore we restrict our attention to decomposition over finite fields and over the algebraic or real closure of number fields. The module isomorphism problem can be reformulated as follows. Let A 1 ; : : : ; Am and A 0 1 ; : : : ; A 0 m be two families of n\Thetanmatrices with entries from the field K. The task is to find a nonsingular n\Thetanmatrix X with entries from K such that XA i X \Gamma1 = A 0 i for all 1 i m (if such a matrix exists). In the case when K is the field of the real algebraic numbers, we propose a method for the variant where the matrix X is required to be orthogonal. St. Petersburg Institute for Informatics and Automation of th...
An algorithm to construct representations of finite groups
, 2003
"... Let G be a finite group. It is easy to compute the character of G corresponding to a given complex representation, but much more difficult to compute a representation affording a given character. In part this is due to the fact that a class of equivalent representations contains no natural canonical ..."
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Let G be a finite group. It is easy to compute the character of G corresponding to a given complex representation, but much more difficult to compute a representation affording a given character. In part this is due to the fact that a class of equivalent representations contains no natural canonical representation. Although there is a large literature devoted to computing representations, and methods are known for particular classes of groups, we know of no general method which has been proposed which is practical for any but small groups. We shall describe an algorithm to compute an irreducible matrix representation R which affords a given character χ of a given group G. The algorithm uses properties of the structure of G which can be computed efficiently by a program such as GAP, theoretical results from representation theory, theorems from group theory (including the Classification of Finite Simple Groups), and linear algebra. All results in this paper have been implemented in the GAP package REPSN. 1
Constructive recognition of PSL(2, q)
 Trans. Amer. Math. Soc
"... Existing black box and other algorithms for explicitly recognising groups of Lie type over GF(q) have asymptotic running times which are polynomial in q, whereas the input size involves only log q. This has represented a serious obstruction to ecient recognition of such groups. ..."
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Existing black box and other algorithms for explicitly recognising groups of Lie type over GF(q) have asymptotic running times which are polynomial in q, whereas the input size involves only log q. This has represented a serious obstruction to ecient recognition of such groups.